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We show that the gravitational potential in the plane of an axisymmetricalflat disk where the surface density varies as a power of the radius obeys aninhomogeneous first-order Ordinary Differential Equation (ODE) solvable bystandard techniques. The potential being exactly known at the origin for anypower index (and at infinity as well), the search for solutions consists of aTwo-point Boundary Value Problem (TBVP) with Dirichlet conditions. Thecomputating time is then linear with the number of grid points, instead ofquadratic from direct summation methods. Complex mass distributions which canbe decomposed into a mixture of power law surface density profiles are easilyaccessible through the superposition principle. This ODE definitively takes theplace of the untractable bidimensional Poisson equation for planarcalculations. It opens new horizons to investigate various aspects related toself-gravity in astrophysical disks (force calculations, stability analysis,etc.).Comment: 5 pages, 1 figure, accepted for publication in A&

A new equation for the mid-plane potential of power law disks